Automorphism group of a group: Difference between revisions

Revision as of 03:08, 28 May 2013

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Definition

Symbol-free definition

The automorphism group of a group is defined as a group whose elements are all the automorphisms of the base group, and where the group operation is composition of automorphisms. In other words, it gets a group structure as a subgroup of the group of all permutations of the group.

Definition with symbols

The automorphism group of a group $G$ , denoted $\operatorname {Aut} (G)$ , is a set whose elements are automorphisms $\sigma :G\to G$ , and where the group multiplication is composition of automorphisms. In other words, its group structure is obtained as a subgroup of $\operatorname {Sym} (G)$ , the group of all permutations on $G$ .

Subgroups

Every group-closed automorphism property gives rise to a normal subgroup of the automorphism group. Some of the most important examples are given below:

Group-closed automorphism property	Meaning	Corresponding normal subgroup of the automorphism group
inner automorphism	can be expressed as conjugation by an element of the group, i.e., there exists $g\in G$ such that the map has the form $x\mapsto gxg^{-1}$	it is called the inner automorphism group and is isomorphic to the quotient group $G/Z(G)$ where $Z(G)$ is the center. See group acts as automorphisms by conjugation.
class-preserving automorphism	sends every element to within its automorphism class	the class-preserving automorphism group
IA-automorphism	sends every coset of the derived subgroup to itself, or equivalently, induces the identity map on the abelianization.	the IA-automorphism group
center-fixing automorphism	fixes every element of the center	the center-fixing automorphism group
monomial automorphism	can be expressed using a monomial formula	the momomial automorphism group
normal automorphism	sends every normal subgroup to itself	the normal automorphism group

Facts

Extensible equals inner: An automorphism of a group has the property that it can be extended to an automorphism for any bigger group containing it if and only if the automorphism is an inner automorphism.
Quotient-pullbackable equals inner: An automorphism of a group has the property that it can be pulled back to an automorphism for any group admitting it as a quotient, if and only if the automorphism is an inner automorphism.

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 ===Definition with symbols===
-The '''automorphism group''' of a [[group]] <math>G</math>, denoted <math>Aut(G)</math>, is a set whose elements are automorphisms <math>\sigma:G \to G</math>, and where the group multiplication is composition of automorphisms. In other words, its group structure is obtained as a subgroup of <math>\operatorname{Sym}(G)</math>, the group of all permutations on <math>G</math>.
+The '''automorphism group''' of a [[group]] <math>G</math>, denoted <math>\operatorname{Aut}(G)</math>, is a set whose elements are automorphisms <math>\sigma:G \to G</math>, and where the group multiplication is composition of automorphisms. In other words, its group structure is obtained as a subgroup of <math>\operatorname{Sym}(G)</math>, the group of all permutations on <math>G</math>.
 ==Subgroups==
-Every [[group-closed automorphism property]] gives rise to a [[normal subgroup]] of the automorphism group. Examples are the property of being an [[inner automorphism]], [[class automorphism]], [[extensible automorphism]].
+Every [[group-closed automorphism property]] gives rise to a [[normal subgroup]] of the automorphism group. Some of the most important examples are given below:
+{| class="sortable" border="1"
+! Group-closed automorphism property !! Meaning !! Corresponding normal subgroup of the automorphism group
+|-
+| [[inner automorphism]] || can be expressed as [[conjugation]] by an element of the group, i.e., there exists <math>g \in G</math> such that the map has the form <math>x \mapsto gxg^{-1}</math> || it is called the [[inner automorphism group]] and is isomorphic to the [[quotient group]] <math>G/Z(G)</math> where <math>Z(G)</math> is the [[center]]. See [[group acts as automorphisms by conjugation]].
+|-
+| [[class-preserving automorphism]] || sends every element to within its automorphism class || the class-preserving automorphism group
+|-
+| [[IA-automorphism]] || sends every coset of the [[derived subgroup]] to itself, or equivalently, induces the identity map on the [[abelianization]]. || the IA-automorphism group
+|-
+| [[center-fixing automorphism]] || fixes every element of the center || the center-fixing automorphism group
+|-
+| [[monomial automorphism]] || can be expressed using a monomial formula || the momomial automorphism group
+|-
+| [[normal automorphism]] || sends every normal subgroup to itself || the normal automorphism group
+|}
+==Facts==
+* [[Extensible equals inner]]: An automorphism of a group has the property that it can be extended to an automorphism for any bigger group containing it if and only if the automorphism is an [[inner automorphism]].
+* [[Quotient-pullbackable equals inner]]: An automorphism of a group has the property that it can be pulled back to an automorphism for any group admitting it as a quotient, if and only if the automorphism is an inner automorphism.